Matrix versions of the Hellinger distance

Article Type

Research Article

Publication Title

Letters in Mathematical Physics


On the space of positive definite matrices, we consider distance functions of the form d(A, B) = [tr A(A, B) - tr G(A, B)] 1 / 2, where A(A, B) is the arithmetic mean and G(A, B) is one of the different versions of the geometric mean. When G(A, B) = A1 / 2B1 / 2 this distance is ‖ A1 / 2- B1 / 2‖ 2, and when G(A,B)=(A1/2BA1/2)1/2 it is the Bures–Wasserstein metric. We study two other cases: G(A,B)=A1/2(A-1/2BA-1/2)1/2A1/2, the Pusz–Woronowicz geometric mean, and G(A,B)=exp(logA+logB2), the log Euclidean mean. With these choices, d(A, B) is no longer a metric, but it turns out that d2(A, B) is a divergence. We establish some (strict) convexity properties of these divergences. We obtain characterisations of barycentres of m positive definite matrices with respect to these distance measures. One of these leads to a new interpretation of a power mean introduced by Lim and Palfia, as a barycentre. The other uncovers interesting relations between the log Euclidean mean and relative entropy.

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Open Access, Green